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%I #12 Feb 28 2017 09:28:12
%S 1,1,1,1,2,2,3,1,2,2,2,3,3,4,1,1,5,2,5,3,2,2,4,4,2,3,4,4,6,4,3,1,4,5,
%T 5,4,4,5,23,5,4,5,22,6,2,2,24,6,25,3,3,4,23,3,21,5,2,3,21,3,25,26,21,
%U 1,6,24,20,25,23,22,27,4,26,27,36,28,35,22,33,5,30,31,20,25,28,29,20,26,29
%N Function f(x) = EulerPhi(x) + floor(x/2) is iterated; a(n) is the length of transient part and terminal cycle if the iteration was initiated at n. So a(n) is the number of distinct terms arising during iteration.
%C Observation (see also _Labos Elemer_'s comment at A097026): most n < 1000 have 1 <= a(n) <= 108; the following have a(n) > 1000 if finite: {163, 182, 196, 243, 283, 331, 423, 487, 495, 503, 511, 523, 533, 551, 559, 571, 583, 591, 593, 606, 611, 623, 642, 646, 651, 679, 685, 687, 725, 726, 729, 731, 732, 745, 746, 753, 755, 757, 758, 767, 779, 781, 783, 791, 799, 809, 811, 814, 839, 850, 855, 857, 859, 867, 869, 871, 875, 876, 885, 886, 888, 891, 895, 906, 908, 911, 913, 914, 915, 916, 921, 922, 923, 931, 937, 942, 959, 962, 964, 970, 971, 977, 985, 991}. - _Michael De Vlieger_, Feb 27 2017
%F a(n) = A097026(n) + A097027(n) = c(n) + t(n).
%e For n=70, iteration list = {70, 59, 87, 99, 109, 162, 135, 139, 207, 235, 301, 402, 333, 382, 381, 442, [413, 554, 553, 744, 612, 498], 413}, a(70) = 22.
%e n=2^j: a(2^j)=1, powers of 2 are fixed points, free of transients, so t + c = 0 + 1 = 1.
%t Table[Length@ Union@ NestList[EulerPhi@ # + Floor[#/2] &, n, 10^3], {n, 10^3}] (* _Michael De Vlieger_, Feb 27 2017 *)
%Y Cf. A000010, A097026, A097028, A097029.
%K nonn
%O 1,5
%A _Labos Elemer_, Aug 27 2004
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